Hyperelliptic Schottky Problem and Stable Modular Forms

It is well known that, fixed an even, unimodular, positive definite quadratic form, one can construct a modular form in each genus; this form is called the theta series associated to the quadratic form. Varying the quadratic form, one obtains the ring of stable modular forms. We show that the differences of theta series associated to specific pairs of quadratic forms vanish on the locus of hyperelliptic Jacobians in each genus. In our examples, the quadratic forms have rank 24, 32 and 48. The proof relies on a geometric result about the boundary of the Satake compactification of the hyperelliptic locus. We also study the monoid formed by the moduli space of all principally polarised abelian varieties, the operation being the product of abelian varieties. We use this construction to show that the ideal of stable modular forms vanishing on the hyperelliptic locus in each genus is generated by differences of theta series.Comment: Final version, title changed, published in Documenta Mathematic

Jacobian Nullwerte, Periods and Symmetric Equations for Hyperelliptic Curves

We propose a solution to the hyperelliptic Schottky problem, based on the use of Jacobian Nullwerte and symmetric models for hyperelliptic curves. Both ingredients are interesting on its own, since the first provide period matrices which can be geometrically described, and the second have remarkable arithmetic properties.Comment: To appear in "Annales de l'Institut Fourier

Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group

This paper is devoted to the explicit description of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism group is cyclic of order coprime to the characteristic of their ground field. Along the way, we obtain an arithmetic criterion for the existence of a hyperelliptic descent. The obstruction is described by the so-called arithmetic dihedral invariants of the curves in question. If it vanishes, then the use of these invariants also allows the explicit determination of a model over the field of moduli; if not, then one obtains a hyperelliptic model over a degree 2 extension of this field.Comment: 35 pages; improve the readability of the pape

Canonical curves with low apolarity

Let $k$ be an algebraically closed field and let $C$ be a non--hyperelliptic smooth projective curve of genus $g$ defined over $k$. Since the canonical model of $C$ is arithmetically Gorenstein, Macaulay's theory of inverse systems allows to associate to $C$ a cubic form $f$ in the divided power $k$--algebra $R$ in $g-2$ variables. The apolarity of $C$ is the minimal number $t$ of linear form in $R$ needed to write $f$ as sum of their divided power cubes. It is easy to see that the apolarity of $C$ is at least $g-2$ and P. De Poi and F. Zucconi classified curves with apolarity $g-2$ when $k$ is the complex field. In this paper, we give a complete, characteristic free, classification of curves $C$ with apolarity $g-1$ (and $g-2$)